A325267 - OEIS

%I #4 Apr 18 2019 16:54:57

%S 0,0,1,1,3,5,7,12,17,24,33,44,57,76,100,129,168,214,282,355,462,586,

%T 755,937,1202,1493,1900,2349,2944,3621,4520,5514,6813,8298,10150,

%U 12240,14918,17931,21654,25917,31081,37029,44256,52474,62405,73724,87378,102887

%N Number of integer partitions of n with omicron 2.

%C The Heinz numbers of these partitions are given by A304634.

%C The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.

%e The a(1) = 1 through a(8) = 17 partitions:

%e   (11)  (21)  (22)   (32)    (33)     (43)      (44)

%e               (31)   (41)    (42)     (52)      (53)

%e               (211)  (221)   (51)     (61)      (62)

%e                      (311)   (411)    (322)     (71)

%e                      (2111)  (2211)   (331)     (332)

%e                              (3111)   (511)     (422)

%e                              (21111)  (2221)    (611)

%e                                       (3211)    (3221)

%e                                       (4111)    (3311)

%e                                       (22111)   (4211)

%e                                       (31111)   (5111)

%e                                       (211111)  (22211)

%e                                                 (32111)

%e                                                 (41111)

%e                                                 (221111)

%e                                                 (311111)

%e                                                 (2111111)

%t Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{_},1,_,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==2&]],{n,0,30}]

%Y Cf. A056239, A112798, A181819, A304634, A304636, A323014, A323023, A325250, A325273, A325277.

%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

%K nonn

%O 0,5

%A _Gus Wiseman_, Apr 18 2019